Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-x+3y &= 6 \\ -3x-9y &= 4\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $1$ $\begin{align*}-3x+9y &= 18\\ -3x-9y &= 4\end{align*}$ Add the top and bottom equations. $-6x = 22$ Divide both sides by $-6$ and reduce as necessary. $x = -\dfrac{11}{3}$ Substitute $-\dfrac{11}{3}$ for $x$ in the top equation. $+ \dfrac{11}{3}+3y = 6$ $\dfrac{11}{3}+3y = 6$ $3y = \dfrac{7}{3}$ $y = \dfrac{7}{9}$ The solution is $\enspace x = -\dfrac{11}{3}, \enspace y = \dfrac{7}{9}$.